Completeness in Differential Approximation Classes

  • G. Ausiello
  • C. Bazgan
  • M. Demange
  • V. Th. Paschos
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)


We study completeness in differential approximability classes. In differential approximation, the quality of an approximation algorithm is the measure of both how far is the solution computed from a worst one and how close is it to an optimal one. The main classes considered are DAPX, the differential counterpart of APX, including the NP optimization problems approximable in polynomial time within constant differential approximation ratio and the DGLO, the differential counterpart of GLO, including problems for which their local optima guarantee constant differential approximation ratio. We define natural approximation preserving reductions and prove completeness results for the class of the NP optimization problems (class NPO), as well as for DAPX and for a natural subclass of DGLO. We also define class 0-APX of the NPO problems that are not differentially approximable within any ratio strictly greater than 0 unless P = NP. This class is very natural for differential approximation, although has no sense for the standard one. Finally, we prove the existence of hard problems for a subclass of DPTAS, the differential counterpart of PTAS, the class of NPO problems solvable by polynomial time differential approximation schemata.


Polynomial Time Approximation Ratio Approximation Class Polynomial Time Approximation Schema Approximation Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Orponen, P., Mannila, H.: On approximation preserving reductions: complete problems and robust measures. Technical Report C-1987-28, Dept. of Computer Science, University of Helsinki, Finland (1987) Google Scholar
  2. 2.
    Crescenzi, P., Panconesi, A.: Completeness in approximation classes. Inform. and Comput. 93, 241–262 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Ausiello, G., Crescenzi, P., Protasi, M.: Approximate solutions of NP optimization problems. Theoret. Comput. Sci. 150, 1–55 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Crescenzi, P., Trevisan, L.: On approximation scheme preserving reducibility and its applications. In: Thiagarajan, P.S. (ed.) FSTTCS 1994. LNCS, vol. 880, pp. 330–341. Springer, Heidelberg (1994)Google Scholar
  6. 6.
    Ausiello, G., D’Atri, A., Protasi, M.: On the structure of combinatorial problems and structure preserving reductions. In: Salomaa, A., Steinby, M. (eds.) ICALP 1977. LNCS, vol. 52. Springer, Heidelberg (1977)Google Scholar
  7. 7.
    Demange, M., Paschos, V.T.: On an approximation measure founded on the links between optimization and polynomial approximation theory. Theoret. Comput. Sci. 158, 117–141 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Ausiello, G., Protasi, M.: NP optimization problems and local optima graph theory. In: Alavi, Y., Schwenk, A. (eds.) Combinatorics and applications. Proc. 7th Quadriennal International Conference on the Theory and Applications of Graphs, vol. 2, pp. 957–975 (1995)Google Scholar
  9. 9.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and approximation. Combinatorial optimization problems and their approximability properties. Springer, Berlin (1999)zbMATHGoogle Scholar
  10. 10.
    Ausiello, G., Bazgan, C., Demange, M., Paschos, V.T.: Completeness in differential approximation classes. Cahier du LAMSADE 204, LAMSADE, Université Paris-Dauphine (2003), Available on
  11. 11.
    Crescenzi, P., Kann, V., Silvestri, R., Trevisan, L.: Structure in approximation classes. SIAM J. Comput. 28, 1759–1782 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Monnot, J.: Differential approximation results for the traveling salesman and related problems. Inform. Process. Lett. 82, 229–235 (2002)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Hassin, R., Khuller, S.: z-approximations. J. Algorithms 41, 429–442 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Bazgan, C., Paschos, V.T.: Differential approximation for optimal satisfiability and related problems. European J. Oper. Res. 147, 397–404 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Toulouse, S.: Approximation polynomiale: optima locaux et rapport différentiel. PhD thesis, LAMSADE, Université Paris-Dauphine (2001) Google Scholar
  16. 16.
    Ausiello, G., Protasi, M.: Local search, reducibility and approximability of NPoptimization problems. Inform. Process. Lett. 54, 73–79 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Monnot, J., Paschos, V.T., Toulouse, S.: Optima locaux garantis pour l’approximation differéntielle. Technical Report 203, LAMSADE, Université Paris-Dauphine (2002), Available on
  18. 18.
    Monnot, J., Paschos, V.T., Toulouse, S.: Approximation algorithms for the traveling salesman problem. Mathematical Methods of Operations Research 57, 387–405 (2003)MathSciNetGoogle Scholar
  19. 19.
    Khanna, S., Motwani, R., Sudan, M., Vazirani, U.: On syntactic versus computational views of approximability. SIAM J. Comput. 28, 164–191 (1998)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • G. Ausiello
    • 1
  • C. Bazgan
    • 2
  • M. Demange
    • 3
  • V. Th. Paschos
    • 2
  1. 1.Dipartimento di Informatica e SistemisticaUniversità degli Studi di Roma “La Sapienza” 
  2. 2.LAMSADEUniversité Paris-Dauphine 
  3. 3.Department of Decision and Information SystemsESSEC 

Personalised recommendations